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Register a new userPropagation of exponential phase space singularities for Schrodinger equations with quadratic Hamiltonians
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We study propagation of phase space singularities for the initial value Cauchy problem for a class of Schrodinger equations. The Hamiltonian is the Weyl quantization of a quadratic form whose real part is non-negative. The equations are studied in the framework of projective Gelfand--Shilov spaces and their distribution duals. The corresponding notion of singularities is called the Gelfand--Shilov wave front set and means the lack of exponential decay in open cones in phase space. Our main result shows that the propagation is determined by the singular space of the quadratic form, just as in the framework of the Schwartz space, where the notion of singularity is the Gabor wave front set.
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We revisit the paper [Mel86] by R. Melrose, providing a full proof of the main theorem on propagation of singularities for subelliptic wave equations, and linking this result with sub-Riemannian geometry. This result asserts that singularities of subelliptic wave equations only propagate along null-bicharacteristics and abnormal extremal lifts of singular curve. As a new consequence, for x = y and denoting by K G the wave kernel, we obtain that the singular support of the distribution t $rightarrow$ K G (t, x, y) is included in the set of lengths of the normal geodesics joining x and y, at least up to the time equal to the minimal length of a singular curve joining x and y.
We investigate the invariance of the Gibbs measure for the fractional Schrodinger equation of exponential type (expNLS) $ipartial_t u + (-Delta)^{frac{alpha}2} u = 2gammabeta e^{beta|u|^2}u$ on $d$-dimensional compact Riemannian manifolds $mathcal{M}$, for a dispersion parameter $alpha>d$, some coupling constant $beta>0$, and $gamma eq 0$. (i) We first study the construction of the Gibbs measure for (expNLS). We prove that in the defocusing case $gamma>0$, the measure is well-defined in the whole regime $alpha>d$ and $beta>0$ (Theorem 1.1 (i)), while in the focusing case $gamma<0$ its partition function is always infinite for any $alpha>d$ and $beta>0$, even with a mass cut-off of arbitrary small size (Theorem 1.1 (ii)). (ii) We then study the dynamics (expNLS) with random initial data of low regularity. We first use a compactness argument to prove weak invariance of the Gibbs measure in the whole regime $alpha>d$ and $0<beta < beta^star_alpha$ for some natural parameter $0<beta^star_alphasim (alpha-d)$ (Theorem 1.3 (i)). In the large dispersion regime $alpha>2d$, we can improve this result by constructing a local deterministic flow for (expNLS) for any $beta>0$. Using the Gibbs measure, we prove that solutions are almost surely global for $0<beta llbeta^star_alpha$, and that the Gibbs measure is invariant (Theorem 1.3 (ii)). (iii) Finally, in the particular case $d=1$ and $mathcal{M}=mathbb{T}$, we are able to exploit some probabilistic multilinear smoothing effects to build a probabilistic flow for (expNLS) for $1+frac{sqrt{2}}2<alpha leq 2$, locally for arbitrary $beta>0$ and globally for $0<beta ll beta^star_alpha$ (Theorem 1.5).
This paper deals with the 2-D Schrodinger equation with time-oscillating exponential nonlinearity $ipartial_t u+Delta u= theta(omega t)big(e^{4pi|u|^2}-1big)$, where $theta$ is a periodic $C^1$-function. We prove that for a class of initial data $u_0 in H^1(mathbb{R}^2)$, the solution $u_{omega}$ converges, as $|omega|$ tends to infinity to the solution $U$ of the limiting equation $ipartial_t U+Delta U= I(theta)big(e^{4pi|U|^2}-1big)$ with the same initial data, where $I(theta)$ is the average of $theta$.
We study a nonlinear equation in the half-space ${x_1>0}$ with a Hardy potential, specifically [-Delta u -frac{mu}{x_1^2}u+u^p=0quadtext{in}quad mathbb R^n_+,] where $p>1$ and $-infty<mu<1/4$. The admissible boundary behavior of the positive solutions is either $O(x_1^{-2/(p-1)})$ as $x_1to 0$, or is determined by the solutions of the linear problem $-Delta h -frac{mu}{x_1^2}h=0$. In the first part we study in full detail the separable solutions of the linear equations for the whole range of $mu$. In the second part, by means of sub and supersolutions we construct separable solutions of the nonlinear problem which behave like $O(x_1^{-2/(p-1)})$ near the origin and which, away from the origin have exactly the same asymptotic behavior as the separable solutions of the linear problem. In the last part we construct solutions that behave like $O(x_1^{-2/(p-1)})$ at some prescribed parts of the boundary, while at the rest of the boundary the solutions decay or blowup at a slower rate determined by the linear part of the equation.
The main purpose of this paper is to study the global propagation of singularities of viscosity solution to discounted Hamilton-Jacobi equation begin{equation}label{eq:discount 1}tag{HJ$_lambda$} lambda v(x)+H( x, Dv(x) )=0 , quad xin mathbb{R}^n. end{equation} We reduce the problem for equation eqref{eq:discount 1} into that for a time-dependent evolutionary Hamilton-Jacobi equation. We proved that the singularities of the viscosity solution of eqref{eq:discount 1} propagate along locally Lipschitz singular characteristics which can extend to $+infty$. We also obtained the homotopy equivalence between the singular set and the complement of associated the Aubry set with respect to the viscosity solution of equation eqref{eq:discount 1}.
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